Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality

$$ \text{Tr} \left( G^2 \right) \leq \text{Tr} \left( GUGU^{-1} \right). $$

If $U$ and $G$ commute, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to

$$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$

and then using the Cauchy-Schwarz inequality. Unfortunately, I have not found a solution yet.